Question

the following information pertains to three shipping terminals operated by a certain corporation. their internal auditor randomly selects one set of shipping documents, ascertaining that the set selected contains an error. find the probability that the error occurred in the sea terminal. which of the following gives the probability that the error occurred in the sea terminal? a. 0.07 b. 0.36 c. 1.38 d. 0.23
terminal | percentage of cargo handled | percent error
land | 60 | 5
air | 30 | 9
sea | 10 | 17

Explanation:

Step1: Define events and given values

Let $L$ = Land Terminal, $A$ = Air Terminal, $S$ = Sea Terminal, $E$ = Error.
Given:
$P(L) = 0.60$, $P(A) = 0.30$, $P(S) = 0.10$
$P(E|L) = 0.05$, $P(E|A) = 0.09$, $P(E|S) = 0.17$

Step2: Calculate total probability of error

Use the law of total probability:
$$P(E) = P(E|L)P(L) + P(E|A)P(A) + P(E|S)P(S)$$
$$P(E) = (0.05 \times 0.60) + (0.09 \times 0.30) + (0.17 \times 0.10)$$
$$P(E) = 0.03 + 0.027 + 0.017 = 0.074$$

Step3: Apply Bayes' Theorem

Find $P(S|E)$:
$$P(S|E) = \frac{P(E|S)P(S)}{P(E)}$$
$$P(S|E) = \frac{0.17 \times 0.10}{0.074} = \frac{0.017}{0.074} \approx 0.23$$

Answer:

D. 0.23